 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013-resit/9/a/solution

Take laboratory displacements from the equilibrium positions with the support at

f = 0

. The left spring's dynamic extension is

x_{1} - f (t)

and the spring between the masses has dynamic extension

x_{2} - x_{1}

m \overset{x}{¨}_{1} = - k (x_{1} - f) + 2 k (x_{2} - x_{1}), 2 m \overset{x}{¨}_{2} = - 2 k (x_{2} - x_{1}) .

Equivalently,

m (1002) \ddot{x} + k (3 - 2 - 2 2) x = k (f (t) 0) .

The natural length sets the equilibrium separations but disappears from these displacement equations. The support is externally prescribed, so no additional support equation is required.

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